Primary tabs

subgroups with coprime orders

Proof. Let GGG and HHH be such subgroups and |G|G|G| and |H|H|H| their orders. Then the intersectionG∩HGHG\!\cap\!H is a subgroup of both GGG and HHH. By Lagrange’s theorem, |G∩H|GH|G\!\cap\!H|divides both |G|G|G| and |H|H|H| and consequently it divides also gcd⁡(|G|,|H|)GH\gcd(|G|,\,|H|) which is 1. Therefore |G∩H|=1GH1|G\!\cap\!H|=1, whence the intersection contains only the identity element.